OR Spectrum

, Volume 36, Issue 2, pp 281–296 | Cite as

Solving elementary shortest-path problems as mixed-integer programs

Regular Article

Abstract

Ibrahim et al. (Int Trans Oper Res 16:361–369, 2009) presented and analyzed two integer programming formulations for the elementary shortest-path problem (ESPP), which is known to be NP-hard if the underlying digraph contains negative cycles. In fact, the authors showed that a formulation based on multi-commodity flows possesses a significantly stronger LP relaxation than a formulation based on arc flow variables. Since the ESPP is essentially an integer problem, the contribution of our paper lies in extending this research by comparing the formulations with regard to the computation time and memory requirements required for their integer solution. Moreover, we assess the quality of the lower bounds provided by an integer relaxation of the multi-commodity flow formulation.

Keywords

Elementary shortest-path problem Negative cycles Mixed-integer programming 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ahuja R, Magnanti T, Orlin J (1993) Network flows. Prentice-Hall, Upper Saddle RiverGoogle Scholar
  2. Aráoz J, Fernández E, Meza O (2009) Solving the prize-collecting rural postman problem. Eur J Oper Res 196(3): 886–896. doi:10.1016/j.ejor.2008.04.037 CrossRefGoogle Scholar
  3. Baldacci R, Mingozzi A, Roberti R (2012) Recent exact algorithms for solving the vehicle routing problem under capacity and time window constraints. Eur J Oper Res 218: 1–6CrossRefGoogle Scholar
  4. Boland N, Dethridge J, Dumitrescu I (2006) Accelerated label setting algorithms for the elementary resource constrained shortest path problem. Oper Res Lett 34(1): 58–68CrossRefGoogle Scholar
  5. Boost (2012) Boost graph library. http://www.boost.org
  6. Crainic T, Ricciardi N, Storchi G (2009) Models for evaluating and planning city logistics systems. Transp Sci 43(4): 432–454. doi:10.1287/trsc.1090.0279 CrossRefGoogle Scholar
  7. Desaulniers G, Desrosiers J, Ioachim I, Solomon M, Soumis F, Villeneuve D (1998) A unified framework for deterministic time constrained vehicle routing and crew scheduling problems. In: Crainic T, Laporte G (eds) Fleet management and logistics. Kluwer, Boston, pp 57–93Google Scholar
  8. Drexl M (2012) Synchronization in vehicle routing—a survey of VRPs with multiple synchronization constraints. Transp Sci. doi:10.1287/trsc.1110.0400
  9. Feillet D, Dejax P, Gendreau M (2005) Traveling salesman problems with profits. Transp Sci 39(2): 188–205. doi:10.1287/trsc.1030.0079 CrossRefGoogle Scholar
  10. Fukasawa R, Longo H, Lysgaard J, Poggide Arago M, Reis M, Uchoa E, Werneck RF (2006) Robust branch-and-cut-and-price for the capacitated vehicle routing problem. Math Program Ser A 106(3): 491–511CrossRefGoogle Scholar
  11. Golden B, Raghavan S, Wasil E (eds) (2008) The vehicle routing problem: latest advances and new challenges. Operations research/computer science interfaces series, vol 43. Springer, BerlinGoogle Scholar
  12. Gutin G, Punnen A (eds) (2002) The traveling salesman problem and its variations. Kluwer, DordrechtGoogle Scholar
  13. Ibrahim M, Maculan N, Minoux M (2009) A strong flow-based formulation for the shortest path problem in digraphs with negative cycles. Int Trans Oper Res 16(3): 361–369. doi:10.1111/j.1475-3995.2008.00681.x CrossRefGoogle Scholar
  14. Irnich S, Desaulniers G (2005) Shortest path problems with resource constraints. In: Desaulniers G, Desrosiers J, Solomon M (eds) Column generation. Springer, New York, pp 33–65Google Scholar
  15. Irnich S, Villeneuve D (2006) The shortest path problem with resource constraints and k-cycle elimination for k ≥  3. INFORMS J Comput 18(3): 391–406CrossRefGoogle Scholar
  16. Jepsen M, Petersen B, Spoorendonk S (2008) A branch-and-cut algorithm for the elementary shortest path problem with a capacity constraint. Technical report 08/01, Department of Computer Science, University of CopenhagenGoogle Scholar
  17. Righini G, Salani M (2006) Symmetry helps: bounded bi-directional dynamic programming for the elementary shortest path problem with resource constraints. Discrete Optim 3(3): 255–273. doi:10.1016/j.disopt.2006.05.007 CrossRefGoogle Scholar
  18. Skorobohatyj G (1999) Finding a minimum cut between all pairs of nodes in an undirected graph. http://elib.zib.de/pub/Packages/mathprog/mincut/all-pairs/index.html. Accessed 25 April 2012
  19. Toth P, Vigo D (eds) (2002) The vehicle routing problem. SIAM Monographs on Discrete Mathematics and Applications, PhiladelphiaGoogle Scholar
  20. Wayne K (2008) Union-find algorithms. http://www.cs.princeton.edu/~rs/AlgsDS07/01UnionFind.pdf. Accessed 25 April 2012

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Chair of Logistics Management, Gutenberg School of Management and EconomicsJohannes Gutenberg UniversityMainzGermany
  2. 2.Fraunhofer Centre for Applied Research on Supply Chain Services (SCS)NurembergGermany

Personalised recommendations